Module 5 : Ginzburg-Landau phenomenological
Lecture 7 : Field and order parameter variation inside a vortex
 
Magnetic properties of classic Type-II superconductors
As seen earlier, superconductors with $\kappa>\frac{1}{\sqrt{2}}$ have solutions of the GL equations with non-zero order parameter for filds less than $H_{c2}$. The Abrikosov solution has a regular array of vortices. Each unit cell of the flux line array has total flux equal to $\phi_{0}=\frac{hc}{2e}$. We will now examine the solution to the GL equation at $H_{c1}$ where the first vortex enters the superconductor.
Field and order parameter variation inside a vortex
Consider a situation where the applied field is small such that the vortex density is small. No interaction between vortices is considered. As argued earlier, the term in a magnetic system, analogous to the $-PdV$ term of a hydrostatic system, is $HdM$. Therefore, when $H$ is the control parameter (held constant), the relevant free energy is the Gibbs free energy. At $H=H_{c1}$
 
\begin{displaymath} G_{s}\mid_{no\, flux}=G_{s}\mid_{first\, vortex} \end{displaymath}
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